A minimax approach to duality for linear distributional sensitivity testing

We consider the dual formulation of the problem of finding the maximum of Eν[f(X)], where ν is allowed to vary over all the probability measures on a Polish space X for which dc(μ,ν)≤r, with dc an optimal transport distance, f a real-valued function on X satisfying some regularity, μ a 'baseline' measure and r≥ 0. Whereas some derivations of the dual rely on Fenchel duality, applied on a vector space of functions in duality with a vector space of measures, we impose compactness on X to allow the use of the minimax theorem of Ky Fan, which does not require vector space structure.